Main Idea Idea To study the discretizations of dynamical systems - PowerPoint PPT Presentation

On the Discretization of Nonholonomic Dynamics in R n Fernando Jim´ enez (Joint work with J¨ urgen Scheurle) Technische Universit¨ at M¨ unchen Nonholonomic Mechanics and Optimal Control Workshop Institut Henri Poincar´ e, Paris November, 2014 Institut Henri Poincar´ e (Paris) Nov 2014 1 / 48

Main Idea Idea To study the discretizations of dynamical systems viewed as perturbations: Qualitative and quantitative aspects. Question Can we understand a discretization of NH systems as a non-autonomous perturbation of the continuous dynamics? “On the discretization of nonholonomic dynamics on R n ”, Jim´ enez and Scheurle; Preprint arXiv:1407.2116 (Submitted). Remark We do not consider autonomous perturbations of the continuous dynamics (BEA). Institut Henri Poincar´ e (Paris) Nov 2014 2 / 48

Autonomous Perturbation: Backward Error Analysis ODE: ˙ x = h ( x ) . Numerical flow x k + 1 = φ ( ǫ, x k ) , where x k ≃ x ( t k ) , s.t. t k = t 0 + ǫ k . ǫ time step. φ ( ǫ, x ) = x + ǫ h ( x ) + ǫ 2 d 2 ( x ) + ǫ 3 d 3 ( x ) + ... Order of consistency | x ( t k ) − x k | = O ( ǫ p + 1 ) . BEA: ˙ ˜ x = h ǫ (˜ x ) , s.t. ˜ x ( t k ) = x k . x ) + ǫ 2 h 3 (˜ ˙ ˜ x = h (˜ x ) + ǫ h 2 (˜ x ) + .... Comparing series expansions d 2 ( y ) − 1 2 ! h ′ h ( x ) , h 2 ( y ) = d 3 ( y ) − 1 3 !( h ′′ ( x ) + h ′ h ′ h ( y )) − 1 2 !( h ′ h 2 ( y ) + h ′ h 3 ( y ) = 2 h ( x )) . Drawback: Backward Error Analysis is an Asymptotic Theory Institut Henri Poincar´ e (Paris) Nov 2014 3 / 48

Outline Introduction The Lagrangian nonholonomic problem Discretization of nonholonomic dynamics as perturbation Variational integrators Nonholonomic integrators Examples and plots Institut Henri Poincar´ e (Paris) Nov 2014 4 / 48

Introduction Fielder B and Scheurle J: “Discretization of homoclinic orbits, rapid forcing and invisible chaos.” Memoirs of the American Mathematical Society, 119(570) , (1996). Theorem Suposse that h ∈ C l ( R n , R n ) , l ≥ 1, and consider the autonomous ODE ˙ x = h ( x ) . (1) Let F ( t , x ) be the fundamental solution of (1) satisfying F ( 0 , x ) = 0 , and assume that there are an integer ρ ≥ 1, a continuous function C : [ 0 , ∞ ) → [ 0 , ∞ ) and a one-step difference approximation of step size ǫ x k + 1 = φ ( ǫ, x k ) , ( 0 < ǫ ≤ ǫ 0 ; k ∈ Z ) which is consistent of order p , i.e. | φ ( ǫ, x ) − F ( ǫ, x ) | ≤ C ( | x | ) ǫ p + 1 . Then, there exists a function g ( ǫ, τ, x ) , as smooth as h and periodic in τ of period 1, such that if G ( t , s ; ǫ, x ) , G ( s , s ; ǫ, x ) = x , is the fundamental solution of the non-autonomous, ǫ − periodic ODE x = h ( x ) + ǫ p g ( ǫ, t /ǫ, x ) , ˙ (2) then G ( ǫ, 0 ; ǫ, x ) = φ ( ǫ, x ) , where G ( ǫ, 0 ; ǫ, · ) : R n → R n is the Poincar´ e map (period map) for (2), corresponding to initial time s = 0. Institut Henri Poincar´ e (Paris) Nov 2014 5 / 48

Introduction Theorem Any p − th order discretization x k + 1 = φ ( ǫ, x k ) of the autonomous Ordinary Differential Equations | x ( t k ) − x k | ∼ O ( ǫ p + 1 ) , x = h ( x ) , ˙ can equivalently be viewed as the time ǫ − period map of a suitable ǫ − periodic non-autonomous perturbation of the original ODE x = h ( x ) + ǫ p g ( ǫ, t /ǫ, x ) , ˙ where ǫ is the fixed discretization lenght. This is, if ˜ x ( t ) is the solution of the perturbed equation, then ˜ x ( t k + 1 ) = x k + 1 = φ ( ǫ, x k ) . Institut Henri Poincar´ e (Paris) Nov 2014 6 / 48

Outline The Lagrangian nonholonomic problem 1 Discretization of nonholonomic dynamics as perturbation 2 Variational integrators 3 Nonholonomic integrators 4 Examples and plots 5 Institut Henri Poincar´ e (Paris) Nov 2014 7 / 48

The Lagrangian nonholonomic problem Q is a n − dimensional manifold with local coordinates q i , i = 1 , ..., n . TQ tangent bundle with local coordinates ( q i , ˙ q i ) . D non-integrable constant rank ( n − m ) distribution on Q ( D ⊂ TQ ) . q i = 0, 1 ≤ α ≤ m . Linear constraints µ α i ( q ) ˙ D ◦ = µ α = µ α , µ α independent. i ( q ) dq i , 1 ≤ α ≤ m � � Lagrangian function L : TQ → R . Institut Henri Poincar´ e (Paris) Nov 2014 8 / 48

Lagrange-d’Alembert principle Triple ( Q , D , L ) defines a nonholonomic system. Lagrange-d’Alembert principle: q : I ⊂ R → Q � t 1 δ L ( q ( t ) , ˙ q ( t )) dt = 0 t 0 for all variations δ q ( t ) ∈ D q ( t ) , q ( t 0 ) , q ( t 1 ) fixed, t ∈ [ t 0 , t 1 ] . Nonholonomic equations (DAE) � ∂ L � − ∂ L d ∂ q i = λ α µ α i ( q ) , dt ∂ ˙ q i q i = 0 , µ α i ( q ) ˙ λ α , α = 1 , ..., m , set of Lagrange multipliers. Institut Henri Poincar´ e (Paris) Nov 2014 9 / 48

Examples: The vertical rolling coin Q = R 2 × S 1 × S 1 , q = ( x , y , θ, ϕ ) . x 2 + ˙ θ 2 + 1 2 I ˙ L = 1 y 2 ) + 1 ϕ 2 , 2 m (˙ 2 J ˙ x − R cos ϕ ˙ ˙ θ = 0 , y − R sin ϕ ˙ ˙ θ = 0 . Lagrange-d’Alembert principle: m ¨ x = λ 1 , m ¨ y = λ 2 , J ¨ ϕ = 0 , I ¨ θ = − λ 1 R cos ϕ − λ 2 R sin ϕ. Institut Henri Poincar´ e (Paris) Nov 2014 10 / 48

Examples: The snakeboard Q = SE ( 2 ) × S 1 × S 1 , q = ( x , y , θ, ψ, φ ) . x 2 + ˙ 2 m r 2 ˙ θ 2 + 1 ψ 2 + J 0 ˙ 2 J 0 ˙ ψ ˙ 2 J 1 ˙ L = 1 y 2 ) + 1 θ + 1 φ 2 , 2 m (˙ y − r cos φ ˙ − sin ( θ + φ ) ˙ x + cos ( θ + φ ) ˙ θ = 0 , y + r cos φ ˙ − sin ( θ − φ ) ˙ x + cos ( θ − φ ) ˙ θ = 0 . Lagrange-d’Alembert principle: m ¨ x = − λ 1 sin ( θ + φ ) − λ 2 sin ( θ − φ ) , m ¨ y = λ 1 cos ( θ + φ ) − λ 2 cos ( θ − φ ) , ( − λ 1 r cos φ + λ 2 r cos φ ) / ( m r 2 − J 0 ) , ¨ θ = ( λ 1 r cos φ − λ 2 r cos φ ) / ( m r 2 − J 0 ) , ¨ ψ = J 1 ¨ φ = 0 . Institut Henri Poincar´ e (Paris) Nov 2014 11 / 48

Outline The Lagrangian nonholonomic problem 1 Discretization of nonholonomic dynamics as perturbation 2 Variational integrators 3 Nonholonomic integrators 4 Examples and plots 5 Institut Henri Poincar´ e (Paris) Nov 2014 12 / 48

Question Question Can we understand a discretization of NH systems as a non-autonomous perturbation of the continuous dynamics? Institut Henri Poincar´ e (Paris) Nov 2014 13 / 48

Nonholonomic equations Q = R n . q i = v i . Regularity condition: det � � ∂ 2 L Second order condition ˙ � = 0. ∂ v j ∂ v i Nonholonomic DAE q i = v i , ˙ � ∂ 2 L � − 1 � − ∂ 2 L � ∂ q k ∂ v j v k + ∂ L v i = ∂ q j + λ α µ α ˙ j ( q ) , ∂ v j ∂ v i i ( q ) v i = 0 , µ α ( q ( t 0 ) = q 0 , v ( t 0 ) = v 0 ) , system of 2 n + m Differential Algebraic Equations (together with initial conditions for q ( t ) and v ( t ) ). Local Nonholonomic flow ( q ( t ) , v ( t ) , λ ( t )) = F NH ( q 0 , v 0 ) , s.t. ( q ( t ) , v ( t )) ∈ D . t Institut Henri Poincar´ e (Paris) Nov 2014 14 / 48

Nonholonomic equations x = ( q i , v i ) ∈ R 2 n and λ α ∈ R m , � ∂ 2 L � ( m ij ) = , ∂ v j ∂ v i ϕ : R 2 n → R m , ϕ α ( x ) = µ α i ( q ) v i . Nonholonomic DAE � 0 � ˙ x = f ( x ) + λ α , m − 1 ( x ) ∇ v ϕ α ( x ) ϕ α ( x ) = 0 . � v i , − m ik ∂ q j ∂ v k v j + m ij ∂ L � ∂ 2 L f i q ( x ) , f i � � f ( x ) = v ( x ) = . ∂ q j Institut Henri Poincar´ e (Paris) Nov 2014 15 / 48

Question Question Instead of a DAE, can we obtain an ODE defining the flow within D ? Institut Henri Poincar´ e (Paris) Nov 2014 16 / 48

Constraint submanifold i ( q ) v i = 0 defines a regular submanifold Nonholonomic constraints ϕ ( x ) = µ α D ⊂ R 2 n . Perpendicularity condition: � � 0 �∇ ϕ β ( x ) , f ( x ) + λ α � = 0 , m − 1 ( x ) ∇ v ϕ α ( x ) This determines a unique λ α ( x ) enforcing the vector field to be tangent to D . x = h ( x ) s.t. x ( t ) ∈ D ⊂ R 2 n and h ( x ) ∈ T x D ⊂ T x R 2 n . ˙ Nonholonomic ODE on D ( ϕ ( x ) = 0 ) � � 0 ˙ x = h ( x ) = f ( x ) + λ α ( x ) , m − 1 ( x ) ∇ v ϕ α ( x ) ( x ( t 0 ) = x 0 ∈ D ) . Institut Henri Poincar´ e (Paris) Nov 2014 17 / 48

Result Approach Apply the result by Fielder and Scheurle on this system to obtain a corresponding perturbed ODE on D . Extending that to all of TQ × R m = R 2 n × R m , we obtain Result Any p − th order discretization of the nonholonomic ODE (evolving on D ) may be embedded into a non-autonomous perturbation of the original nonholonomic DAE of the form � � 0 + ǫ p ˜ x = f ( x ) + λ α ˙ g ( ǫ, t /ǫ, x ) , m − 1 ( x ) ∇ v ϕ α ( x ) ϕ α ( x ) = 0 . Institut Henri Poincar´ e (Paris) Nov 2014 18 / 48

Ehresmann connection Bloch, A: “Nonholonomic Mechanics and Control”, Springer-Verlag New-York, (2003). Ehresmann connection: global definition An Ehresmann connection A is a vertical vector-valued one-form on Q , which satisfies A q : T q Q → V q is a linear map at each point q ∈ Q , A is a projection, i.e., A ( v q ) = v q , for all v q ∈ V q . The connection allows T q Q = H q ⊕ V q , s.t. H q = ker A q . Why do we need the connection? It helps to choose convenient coordinates for our system. Moreover, it guarantees the independence of the coordinate choice in TQ . H q = D q := D ∩ T q Q , and thus the constraints are globally expressed as A · v q = 0 for any v q ∈ T q Q . Institut Henri Poincar´ e (Paris) Nov 2014 19 / 48